This Demonstration lets you verify 24 valid syllogisms using Venn diagrams with only one element in the domain. The domain only needs two elements, denoted by "+" and "×", to show that a syllogistic form is not valid.

The universal set is divided into eight subsets by , , and . If a subset is shaded, it is empty. A white subset does not guarantee that it contains an element, but if the sign "+" or "×" is in a subset, then it does have an element. If "+" or "×" is in a shaded subset, there is a contradiction. So the statement that a subset is empty is true if it is shaded, false if either "+" or "×" is in it, and otherwise the statement is undecided.

Snapshots

Details

A monadic formula of first-order logic is one for which all nonlogical symbols are one-place predicates.

Theorem. If is a monadic sentence that is satisfiable, then is true in some interpretation whose domain contains at most members, where is the number of one-place predicate letters and is the number of variables in .

Therefore there is an effective procedure for deciding whether or not a monadic sentence is valid [1, p. 250].

Syllogistic forms are monadic sentences if considered as sentences of the form with predicate letters , , and .